For the bars: Preheat the oven to 350 degrees F. Grease a 13-by-9-inch metal baking pan with nonstick baking spray and line the bottom and sides with parchment paper, leaving an inch of overhang on the sides.
Whisk together the flour, cinnamon, baking powder, baking soda, ginger, salt, nutmeg and cloves in a large bowl until evenly combined. In another bowl, whisk together the granulated sugar, oil, vanilla, eggs and pumpkin puree until smooth. Pour the pumpkin mixture over the dry ingredients and stir until the batter just comes together.
Spread the batter evenly in the prepared pan and bake until a toothpick inserted in the middle of the cake comes out clean, about 40 minutes. Let the cake cool for 30 minutes in the pan, then unmold and let cool completely on a wire rack.
For the icing: Combine the cream cheese and butter in a medium bowl and beat on medium speed with a handheld mixer until smooth. Add the confectioners' sugar, vanilla and salt and beat until evenly combined and smooth. Spread the icing on the top of the cake in a smooth and even layer. Cut the cake into twenty-four 2-inch squares and serve.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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